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Question

Two circles of unit radius touch each other and each of them touches internally a circle of radius 2 units, as shown in the following figure. The radius of the circle which touches all the three circles is
92758.png

A
5
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B
32
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C
23
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D
None of these
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Solution

The correct option is C 23
GivenC&Aarecentresoftwocirclesofradii=1unitandtheytouchatB.Thesetwocirclesinternallytouchanothercirclewhoseradiusis2unitsi.ediameter=2×2units=4units..ThereisathirdcirclewithcentreasOandittouchesallthethreepreviouscircles.Tofindouttheradius=rofthecirclewithcentreO.SolutionBD=2AD=2×1unit=2units.NowBE&BDaresamestraightlinesincewhentwocirclestoucheachotherthenthelinejoiningthecentreswillpassthroughthepointofcontact.ButED=BE+AD=(2+2)units=4unitswhichisthediameteroftheenclosingcircle.NowBTisthecommontangetofthecircleswithcentresA&C.BTED.i.eΔOABisarightonewithhypotenuseasOA=1+r.(OAwillpassthroughPsincethesecirclestouchatP)AlsoBTistheradiusoftheenclosingcircle.i.eBT=2unitsandOB=BTOT=2r.So,applyingPythagorastheoreminΔOAB,wehaveOA2=OB2+AB2(1+r)2=(2r)2+(1)244r=2rr=23unit.AnsOptionC.
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